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Invertibility Issues for Toeplitz Plus Hankel Operators and Their Close Relatives

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Operator Theory, Functional Analysis and Applications

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 282))

Abstract

The paper describes various approaches to the invertibility of Toeplitz plus Hankel operators in Hardy and l p-spaces, integral and difference Wiener-Hopf plus Hankel operators and generalized Toeplitz plus Hankel operators. Special attention is paid to a newly developed method, which allows to establish necessary, sufficient and also necessary and sufficient conditions of invertibility, one-sided and generalized invertibility for wide classes of operators and derive efficient formulas for the corresponding inverses. The work also contains a number of problems whose solution would be of interest in both theoretical and applied contexts.

This work was supported by the Special Project on High-Performance Computing of the National Key R&D Program of China (Grant No. 2016YFB0200604), the National Natural Science Foundation of China (Grant No. 11731006) and the Science Challenge Project of China (Grant No. TZ2018001).

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References

  1. J. Baik, E.M. Rains, Algebraic aspects of increasing subsequences. Duke Math. J. 109, 1–65 (2001)

    Article  MathSciNet  Google Scholar 

  2. E.L. Basor, T. Ehrhardt, On a class of Toeplitz + Hankel operators. New York J. Math. 5, 1–16 (1999)

    MathSciNet  MATH  Google Scholar 

  3. E.L. Basor, T. Ehrhardt, Factorization theory for a class of Toeplitz + Hankel operators. J. Oper. Theory 51, 411–433 (2004)

    MathSciNet  MATH  Google Scholar 

  4. E.L. Basor, T. Ehrhardt, Fredholm and invertibility theory for a special class of Toeplitz + Hankel operators. J. Spectral Theory 3, 171–214 (2013)

    Article  MathSciNet  Google Scholar 

  5. E.L. Basor, T. Ehrhardt, Asymptotic formulas for determinants of a special class of Toeplitz + Hankel matrices, in Large Truncated Toeplitz Matrices, Toeplitz Operators, and Related Topics, vol. 259. The Albrecht Böttcher Anniversary Volume. Operator Theory: Advances and Applications (Birkhäuser, Basel, 2017), pp. 125–154

    Google Scholar 

  6. E. Basor, Y. Chen, T. Ehrhardt, Painlevé V and time-dependent Jacobi polynomials. J. Phys. A 43, 015204, 25 (2010)

    Google Scholar 

  7. A. Böttcher, B. Silbermann, Analysis of Toeplitz Operators, 2nd edn. Springer Monographs in Mathematics (Springer, Berlin, 2006)

    Google Scholar 

  8. A. Böttcher, Y.I. Karlovich, I.M. Spitkovsky, Convolution Operators and Factorization of Almost Periodic Matrix Functions (Birkhäuser, Basel, 2002)

    Google Scholar 

  9. L.P. Castro, A.P. Nolasco, A semi-Fredholm theory for Wiener-Hopf-Hankel operators with piecewise almost periodic Fourier symbols. J. Oper. Theory 62, 3–31 (2009)

    MathSciNet  MATH  Google Scholar 

  10. L.P. Castro, A.S. Silva, Wiener-Hopf and Wiener-Hopf-Hankel operators with piecewise-almost periodic symbols on weighted Lebesgue spaces. Mem. Diff. Equ. Math. Phys. 53, 39–62 (2011)

    MathSciNet  MATH  Google Scholar 

  11. K. Clancey, I. Gohberg, Factorization of Matrix Functions and Singular Integral Operators (Birkhäuser, Basel, 1981)

    Google Scholar 

  12. L.A. Coburn, R.G. Douglas, Translation operators on the half-line. Proc. Nat. Acad. Sci. USA 62, 1010–1013 (1969)

    Article  MathSciNet  Google Scholar 

  13. V.D. Didenko, B. Silbermann, Index calculation for Toeplitz plus Hankel operators with piecewise quasi-continuous generating functions. Bull. London Math. Soc. 45, 633–650 (2013)

    Article  MathSciNet  Google Scholar 

  14. V.D. Didenko, B. Silbermann, The Coburn-Simonenko Theorem for some classes of Wiener–Hopf plus Hankel operators. Publ. de l’Institut Mathématique 96(110), 85–102 (2014)

    MathSciNet  MATH  Google Scholar 

  15. V.D. Didenko, B. Silbermann, Some results on the invertibility of Toeplitz plus Hankel operators. Ann. Acad. Sci. Fenn. Math. 39, 443–461 (2014)

    Article  MathSciNet  Google Scholar 

  16. V.D. Didenko, B. Silbermann, Structure of kernels and cokernels of Toeplitz plus Hankel operators. Integr. Equ. Oper. Theory 80, 1–31 (2014)

    Article  MathSciNet  Google Scholar 

  17. V.D. Didenko, B. Silbermann, Generalized inverses and solution of equations with Toeplitz plus Hankel operators. Bol. Soc. Mat. Mex. 22, 645–667 (2016)

    Article  MathSciNet  Google Scholar 

  18. V.D. Didenko, B. Silbermann, Generalized Toeplitz plus Hankel operators: kernel structure and defect numbers. Compl. Anal. Oper. Theory 10, 1351–1381 (2016)

    Article  MathSciNet  Google Scholar 

  19. V.D. Didenko, B. Silbermann, Invertibility and inverses of Toeplitz plus Hankel operators. J. Oper. Theory 72, 293–307 (2017)

    MathSciNet  MATH  Google Scholar 

  20. V.D. Didenko, B. Silbermann, Kernels of Wiener-Hopf plus Hankel operators with matching generating functions, in Recent Trends in Operator Theory and Partial Differential Equations, vol. 258. The Roland Duduchava Anniversary Volume. Operator Theory: Advances and Applications (Birkhäuser, Basel, 2017), pp. 111–127

    Google Scholar 

  21. V.D. Didenko, B. Silbermann, Kernels of a class of Toeplitz plus Hankel operators with piecewise continuous generating functions, in Contemporary Computational Mathematics – A Celebration of the 80th Birthday of Ian Sloan, ed. by J. Dick, F.Y. Kuo, H. Woźniakowski (eds). (Springer, Cham, 2018), pp. 317–337

    Google Scholar 

  22. V.D. Didenko, B. Silbermann, The invertibility of Toeplitz plus Hankel operators with subordinated operators of even index. Linear Algebra Appl. 578, 425–445 (2019)

    Article  MathSciNet  Google Scholar 

  23. V.D. Didenko, B. Silbermann, Invertibility issues for a class of Wiener-Hopf plus Hankel operators. J. Spectral Theory 11 (2021)

    Google Scholar 

  24. R.V. Duduchava, Wiener-Hopf integral operators with discontinuous symbols. Dokl. Akad. Nauk SSSR 211, 277–280 (1973) (in Russian)

    MathSciNet  Google Scholar 

  25. R.V. Duduchava, Integral operators of convolution type with discontinuous coefficients. Math. Nachr. 79, 75–98 (1977)

    Article  MathSciNet  Google Scholar 

  26. R.V. Duduchava, Integral Equations with Fixed Singularities (BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1979)

    MATH  Google Scholar 

  27. R. Edwards, Fourier Series. A Modern Introduction, vol. 1. Graduate Texts in Mathematics, vol. 85 (Springer, Berlin, 1982)

    Google Scholar 

  28. T. Ehrhardt, Factorization theory for Toeplitz+Hankel operators and singular integral operators with flip. Habilitation Thesis, Technische Universität Chemnitz (2004)

    MATH  Google Scholar 

  29. T. Ehrhardt, Invertibility theory for Toeplitz plus Hankel operators and singular integral operators with flip. J. Funct. Anal. 208, 64–106 (2004)

    Article  MathSciNet  Google Scholar 

  30. P.J. Forrester, N.E. Frankel, Applications and generalizations of Fisher-Hartwig asymptotics. J. Math. Phys. 45, 2003–2028 (2004)

    Article  MathSciNet  Google Scholar 

  31. I.C. Gohberg, I.A. Feldman, Convolution Equations and Projection Methods for Their Solution (American Mathematical Society, Providence, 1974)

    Google Scholar 

  32. I. Gohberg, N. Krupnik, One-Dimensional Linear Singular Integral Equations. I, vol. 53. Operator Theory: Advances and Applications (Birkhäuser Verlag, Basel, 1992)

    Google Scholar 

  33. I. Gohberg, N. Krupnik, One-Dimensional Linear Singular Integral Equations. II, vol. 54. Operator Theory: Advances and Applications (Birkhäuser Verlag, Basel, 1992)

    Google Scholar 

  34. S. Grudsky, A. Rybkin, On positive type initial profiles for the KdV equation. Proc. Am. Math. Soc. 142, 2079–2086 (2014)

    Article  MathSciNet  Google Scholar 

  35. S. Grudsky, A. Rybkin, Soliton theory and Hankel operators. SIAM J. Math. Anal. 47, 2283–2323 (2015)

    Article  MathSciNet  Google Scholar 

  36. S.M. Grudsky, A.V. Rybkin, On the trace-class property of Hankel operators arising in the theory of the Korteweg-de Vries equation. Math. Notes 104, 377–394 (2018)

    Article  MathSciNet  Google Scholar 

  37. P. Junghanns, R. Kaiser, A note on Kalandiya’s method for a crack problem. Appl. Numer. Math. 149, 52–64 (2020)

    Article  MathSciNet  Google Scholar 

  38. N.K. Karapetiants, S.G. Samko, On Fredholm properties of a class of Hankel operators. Math. Nachr. 217, 75–103 (2000)

    Article  MathSciNet  Google Scholar 

  39. N.K. Karapetiants, S.G. Samko, Equations with Involutive Operators (Birkhäuser Boston Inc., Boston, 2001)

    Book  Google Scholar 

  40. V.G. Kravchenko, A.B. Lebre, J.S. Rodríguez, Factorization of singular integral operators with a Carleman shift via factorization of matrix functions: the anticommutative case. Math. Nachr. 280, 1157–1175 (2007)

    Article  MathSciNet  Google Scholar 

  41. V.G. Kravchenko, A.B. Lebre, J.S. Rodríguez, Factorization of singular integral operators with a Carleman backward shift: the case of bounded measurable coefficients. J. Anal. Math. 107, 1–37 (2009)

    Article  MathSciNet  Google Scholar 

  42. N.Y. Krupnik, Banach Algebras with Symbol and Singular Integral Operators, vol. 26. Operator Theory: Advances and Applications (Birkhäuser Verlag, Basel, 1987)

    Google Scholar 

  43. A.B. Lebre, E. Meister, F.S. Teixeira, Some results on the invertibility of Wiener-Hopf-Hankel operators. Z. Anal. Anwend. 11, 57–76 (1992)

    Article  MathSciNet  Google Scholar 

  44. G.S. Litvinchuk, I.M. Spitkovskii, Factorization of Measurable Matrix Functions, vol. 25. Operator Theory: Advances and Applications (Birkhäuser Verlag, Basel, 1987)

    Google Scholar 

  45. E. Meister, F. Penzel, F.-O. Speck, F.S. Teixeira, Two-media scattering problems in a half-space, in Partial Differential Equations with Real Analysis. Dedicated to Robert Pertsch Gilbert on the occasion of his 60th birthday (Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1992), pp. 122–146

    Google Scholar 

  46. E. Meister, F.-O. Speck, F.S. Teixeira, Wiener-Hopf-Hankel operators for some wedge diffraction problems with mixed boundary conditions. J. Integral Equ. Appl. 4, 229–255 (1992)

    Article  MathSciNet  Google Scholar 

  47. V.V. Peller, Hankel Operators and Their Applications. Springer Monographs in Mathematics (Springer, New York, 2003)

    Google Scholar 

  48. S.C. Power, C*-algebras generated by Hankel operators and Toeplitz operators. J. Funct. Anal. 31, 52–68 (1979)

    Article  MathSciNet  Google Scholar 

  49. S. Roch, B. Silbermann, Algebras of convolution operators and their image in the Calkin algebra, vol. 90. Report MATH (Akademie der Wissenschaften der DDR Karl-Weierstrass-Institut für Mathematik, Berlin, 1990)

    Google Scholar 

  50. S. Roch, B. Silbermann, A handy formula for the Fredholm index of Toeplitz plus Hankel operators. Indag. Math. 23, 663–689 (2012)

    Article  MathSciNet  Google Scholar 

  51. S. Roch, P.A. Santos, B. Silbermann, Non-Commutative Gelfand Theories. A Tool-Kit for Operator Theorists and Numerical Analysts. Universitext (Springer, London, 2011)

    Google Scholar 

  52. B. Silbermann, The C ∗ -algebra generated by Toeplitz and Hankel operators with piecewise quasicontinuous symbols. Integr. Equ. Oper. Theory 10, 730–738 (1987)

    Google Scholar 

  53. I.B. Simonenko, Some general questions in the theory of Riemann boundary problem. Math. USSR Izvestiya 2, 1091–1099 (1968)

    Article  Google Scholar 

  54. I.J. Šneı̆berg, Spectral properties of linear operators in interpolation families of Banach spaces. Mat. Issled. 9, 2(32), 214–229 (1974) (in Russian)

    Google Scholar 

  55. I.M. Spitkovskiı̆, The problem of the factorization of measurable matrix-valued functions. Dokl. Akad. Nauk SSSR 227, 576–579 (1976) (in Russian)

    Google Scholar 

  56. F.S. Teixeira, Diffraction by a rectangular wedge: Wiener-Hopf-Hankel formulation. Integr. Equ. Oper. Theory 14, 436–454 (1991)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors express their sincere gratitude to anonymous referees for insightful comments and suggestions that helped to improve the paper.

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Authors and Affiliations

  1. Department of Mathematics, SUSTech International Center for Mathematics, Southern University of Science and Technology, Shenzhen, China

    Victor D. Didenko

  2. Technische Universität Chemnitz, Fakultät für Mathematik, Chemnitz, Germany

    Bernd Silbermann

Authors
  1. Victor D. Didenko
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  2. Bernd Silbermann
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Editors and Affiliations

  1. Instituto Superior Técnico, Universidade de Lisboa, Lisboa, Portugal

    M. Amélia Bastos

  2. Departamento de Matemática, Universidade de Aveiro, Aveiro, Portugal

    Luís Castro

  3. Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Lisboa, Portugal

    Alexei Yu. Karlovich

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Didenko, V.D., Silbermann, B. (2021). Invertibility Issues for Toeplitz Plus Hankel Operators and Their Close Relatives. In: Bastos, M.A., Castro, L., Karlovich, A.Y. (eds) Operator Theory, Functional Analysis and Applications. Operator Theory: Advances and Applications, vol 282. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-51945-2_7

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